Q. What are the domain and range of \(f(x)=2\lvert x-4\rvert\)?

Problem: Find the domain and range of \( f(x)=2\lvert x-4\rvert \)

Step 1 – Identify the function type

The function \( f(x)=2\lvert x-4\rvert \) is an absolute value function. Absolute value expressions are defined for every real number because \(\lvert \cdot\rvert\) accepts any real input.

Step 2 – Determine the domain

There are no operations that restrict \(x\) (no division by zero, no even root of a negative). Therefore the domain is all real numbers.

In interval notation: \(\,(-\infty,\infty)\,\).

Step 3 – Absolute value property for the range

For any real number \(a\), \(\lvert a\rvert\ge 0\). Thus \(\lvert x-4\rvert\ge 0\) for all \(x\). Multiplying by 2 preserves nonnegativity, so \(2\lvert x-4\rvert\ge 0\).

Step 4 – Find the minimum and behavior

The minimum occurs when the inside is zero, at \(x=4\): \(f(4)=2\lvert 4-4\rvert=0\). As \(\lvert x-4\rvert\) increases without bound when \(x\) moves away from 4, so does \(f(x)\).

Answer

Domain: \(\,(-\infty,\infty)\,\)

Range: \(\,[0,\infty)\,\)